Finite Group Databasebeta

← #38768804

Subgroups — #38768804

Order 25472.

620 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263882C212yesmaximal, complemented
2#27164019C212yesmaximal, complemented
3C2×#20263872C211yesmaximal, characteristic, complemented
4#27163921C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C199C211yesmaximal, characteristic
6C2×#15691870C2212yescomplemented
7C2×C4⋊Dic199C2211yescharacteristic, complemented
8C2×C22⋊Dic199—22no—
9C2×C4×Dic199C2211yescharacteristic, complemented
10#20263882—24no—
11#20263879—22no—
12#20263876—22no—
13#20263872—22no—
14H(5,𝔽2)×C199—24no—
15H(5,𝔽2)×C199—22no—
16H(5,𝔽2)×C199C414yescomplemented
17C2×#41×C199—22no—
18C2×#41×C199C2212yes—
19C22×D4×C199—22no—
20C22×D4×C199—22no—
21C22×D4×C199C2211yescharacteristic
22C22×Dic199—21no—
23C22×Dic199—21no—
24#15691870—22no—
25#15691870—22no—
26C2×C199⋊C8—22no—
27C4⋊Dic199—22no—
28C22⋊Dic199—44no—
29C4×Dic199—22no—
30C24×C199—22no—
31C24×C199D412yes—
32#41×C199—28no—
33#41×C199—28no—
34#41×C199—22no—
35#41×C199—22no—
36#41×C199C2×C414yes—
37#41×C199D414yes—
38C2×Q8×C199D412yes—
39C2×D4×C199—24no—
40C2×D4×C199—28no—
41C2×D4×C199—28no—
42C2×D4×C199—28no—
43C2×D4×C199—22no—
44C2×D4×C199—28no—
45C2×D4×C199—22no—
46C2×D4×C199—22no—
47C2×D4×C199C2×C412yes—
48C2×D4×C199D412yes—
49C2×D4×C199D412yes—
50C22×C4×C199—22no—
51C22×C4×C199—22no—
52C22×C4×C199C2311yescharacteristic
53C2×Dic199—22no—
54C2×Dic199—41no—
55C2×Dic199—41no—
56C2×Dic199—42no—
57C2×Dic199—41no—
58C199⋊C8—42no—
59C23×C199—28no—
60C23×C199—22no—
61C23×C199—22no—
62C23×C199—22no—
63C23×C199—42no—
64C23×C199—42no—
65C23×C199—82no—
66C23×C199—82no—
67C23×C199C22⋊C414yes—
68C23×C199C2×D411yescharacteristic
69Q8×C199—22no—
70Q8×C199C22⋊C414yescomplemented
71D4×C199—216no—
72D4×C199—24no—
73D4×C199—24no—
74D4×C199—24no—
75D4×C199—24no—
76D4×C199—22no—
77D4×C199C22⋊C414yescomplemented
78C2×C4×C199—28no—
79C2×C4×C199—22no—
80C2×C4×C199—24no—
81C2×C4×C199—24no—
82C2×C4×C199C22×C411yescharacteristic
83C2×C4×C199C2×D412yes—
84C2×C4×C199C2×D412yes—
85C2×C4×C199C2×D411yescharacteristic
86C22×C199—21no—
87C22×C199—21no—
88C22×C199—24no—
89C22×C199—24no—
90C22×C199—44no—
91C22×C199—42no—
92C22×C199—42no—
93C22×C199—42no—
94C22×C199—42no—
95C22×C199—82no—
96C22×C199—88no—
97C22×C199C2×C22⋊C411yescharacteristic
98C22×C199#12011yescharacteristic
99C22×C199#12011yescharacteristic
100C4×C199—24no—
101C4×C199C2×C22⋊C412yes—
102C4×C199#12012yes—
103Dic199—42no—
104Dic199—81no—
105C2×C199—21no—
106C2×C199—21no—
107C2×C199—44no—
108C2×C199—44no—
109C2×C199#37911yescharacteristic
110C2×C199#45311yescharacteristic
111C2×C199#45511yescharacteristic
112C199#179211yescharacteristic, complemented, Sylow
113#1792—1991nomaximal, Sylow
114C2×H(5,𝔽2)D19911yescharacteristic
115C2×#104—1992no—
116#417—1992no—
117C2×#100—1991no—
118#390—1991no—
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)Dic19914yescomplemented
122C2×#41—22no—
123C2×#41C2×D19912yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D19911yescharacteristic
127C2×M16—1992no—
128C2×C4⋊C4—1991no—
129C2×C22⋊C4—3982no—
130C2×C42—1991no—
131#104—3984no—
132#103—3982no—
133#102—3982no—
134#100—3982no—
135C24—22no—
136C24C22⋊D19912yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×Dic19914yes—
142#41C22⋊D19914yes—
143C2×Q8C22⋊D19912yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—22no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×Dic19912yes—
153C2×D4C22⋊D19912yes—
154C2×D4C22⋊D19912yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—3981no—
158C22×C4—3981no—
159C22×C4C22×D19911yescharacteristic
160M16—3982no—
161M16—3982no—
162C2×C8—3982no—
163C4⋊C4—3982no—
164C22⋊C4—7964no—
165C42—3982no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊Dic19914yes—
175C23C2×C22⋊D19911yescharacteristic
176Q8—22no—
177Q8C22⋊Dic19914yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊Dic19914yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7962no—
192C2×C4—7961no—
193C2×C4—3982no—
194C2×C4C22×Dic19911yescharacteristic
195C2×C4C2×C22⋊D19912yes—
196C2×C4C2×C22⋊D19912yes—
197C2×C4C2×C22⋊D19911yescharacteristic
198C8—7962no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊Dic19911yescharacteristic
211C22#2026392411yescharacteristic
212C22#2026392411yescharacteristic
213C4—24no—
214C4—7962no—
215C4—15921no—
216C4C2×C22⋊Dic19912yes—
217C4#2026392412yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#2716387411yescharacteristic
223C2#2716418611yescharacteristic
224C2#2716419611yescharacteristic
subgroupG/H
centerC22#20263924
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting, prime radicalC2×H(5,𝔽2)×C199C2
socleC22×C199#120
cosocleC22×C4×C199C23
hypercenterC2×H(5,𝔽2)D199
hypocenterC199#1792
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#20263882C2semidirect
2. #27164019C2semidirect
3. C2×#20263872C2semidirect
4. #27163921C2semidirect
6. C2×#15691870C22semidirect
7. C2×C4⋊Dic199C22semidirect
9. C2×C4×Dic199C22semidirect
16. H(5,𝔽2)×C199C4semidirect
70. Q8×C199C22⋊C4semidirect
77. D4×C199C22⋊C4semidirect
112. C199#1792semidirect
121. H(5,𝔽2)Dic199semidirect
177. Q8C22⋊Dic199semidirect
184. D4C22⋊Dic199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#202638822
#271640192
C2×#202638722
#271639212
C2×H(5,𝔽2)×C1992
#1792199